This lemma can be proved by a slight modification of Theorem 3.4 of [1]}. To this end, by setting \(\epsilon = 4\sqrt{\frac{d\log (2n_2)+\log (2/\delta )}{n_2 }}\)  and \(\widetilde{\epsilon }= 4\sqrt{\frac{d\log (2n_1)+\log (2/\delta )}{n_1}}\) , then we know \(\epsilon \le \widetilde{\epsilon }\)  (since \(n_1\gg n_2\) ) and
\(&(2n_1)^d \exp \left(-\frac{\epsilon ^2 n_1}{16}\right) \le (2n_1)^d \exp \left(-\frac{\widetilde{\epsilon }^2 n_1}{16}\right) = \delta /2,\\&(2n_2)^d \exp \left(-\frac{\epsilon ^2 n_2}{16}\right) = \delta /2.\) 
